Global anomalies in sigma models with Majorana-Weyl fermions

A bstract We investigate a global sigma model anomaly in two-dimensional sigma models with Majorana-Weyl fermions coupled to a sigma model field with target space G . The anomaly originates from the nontrivial topology of the space of maps and manifests as a phase in the fermion path integral. Using the holonomy formula expressed in terms of the reduced η -invariant, we demonstrate that this anomaly modifies the standard quantization condition of the Wess-Zumino term on G , in close analogy with the three-dimensional parity anomaly. However, our situation is more refined and highlights a qualitatively new phenomenon in two dimensions: whereas in three dimensions the anomalous quantization restricts the level to lie in a half-integer lattice, here it can force the level to take arbitrary real values . Furthermore, our results support the consistency of the low-energy description proposed by Gaiotto, Johnson-Freyd, and Witten for three-dimensional $$ \mathcal{N} $$ N = 1 supersymmetric Yang-Mills theory on an interval, by highlighting a subtle and qualitatively distinct nature of the sigma-model anomalies.